The Reds have struggled at the back this season and their weaknesses in defence were highlighted in their 5-4 win over Norwich on Saturday.
QPR centre-back Steven Caulker has joined the club on loan until the end of the season but manager Jurgen Klopp has admitted they must improve defensively.
Liverpool face Stoke in the second leg of their League Cup semi-final on Tuesday, with Shawcross absent through injury.
The Potters have kept seven clean sheets in their past 13 Premier League games and Thompson says their captain would slot perfectly into the Reds' starting XI.
He told Sky Sports: "When Liverpool evaluate at the end of the season and look at the sort of goals they conceded, I wouldn't be averse to see them looking towards Shawcross as a solution.
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Stoke manager Mark Hughes believes Shawcross is not the only player in his squad who has proved himself to be at least the equal of Liverpool's current crop.
"There should not be any apprehension from the guys," Hughes told a news conference. "They've gone up against Liverpool this year. Most of them have gone up against them in previous seasons as well.
"We've proved we can cause them problems, albeit at Anfield we have not gone that final step and taken wins away with us. But we're really looking forward to it.
"It's not going to be easy. Liverpool at Anfield on any occasion is difficult but in a cup semi-final there is an extra challenge to everybody.
"You could suggest the pressure is on them."